What is known about the higher algebraic K-theory of the complex numbers $\mathbb{C}$?

It's obvious that $K_0(\mathbb{C}) = \mathbb{Z}$. According to Wikipedia, it seems like we should have $K_1(\mathbb{C}) = \mathbb{C}^\times$. It seems like one can write down something for $K_2$, though I don't really understand it very well. Roughly, I want to get a sense of how hard this problem is (blind guess: it is hard).

If it turns out this problem is easy, then the follow-up question is what is the higher algebraic K-theory of $BG$, i.e. the exact category of $G$-representations, for $G$ an algebraic group?


I'm definitely not an expert on algebraic $K$-theory, but here is at least some idea for what is true for $K_i(\mathbf{C})$. The main result is the following:

Theorem (Suslin 1984). Modulo uniquely divisible groups, we have $$K_i(\mathbf{C}) = \begin{cases} 0 & \text{if}\ i\ \text{even}\\ \mathbf{Q}/\mathbf{Z} & \text{if}\ i\ \text{odd} \end{cases}$$

I don't know if you can get a more complete description of the uniquely divisible part. The key idea is to use the following

Theorem (Suslin 1984). $\mathrm{BGL}(\mathbf{C})^+ \to \mathrm{BGL}(\mathbf{C})^\mathrm{top}$ induces isomorphisms on homology and homotopy groups with finite coefficients.

Here, $\mathrm{BGL}(\mathbf{C})$ denotes the classifying space for the discrete group $\mathrm{GL}(\mathbf{C})$, while $\mathrm{BGL}(\mathbf{C})^\mathrm{top} \simeq \mathrm{BU}$ is the classifying space of the topological group $\mathrm{GL}(\mathbf{C})^\mathrm{top}$. The first Theorem then follows by a result due to Weibel; look at the last section in Suslin's paper.

For other references, the relevant chapter (§3) in Weibel's book (I've linked to a draft version) is useful, as is this ICM lecture (§2) by Suslin, and this survey (§22) by Grayson. Note also that Jardine has an alternate proof of the first theorem using sheaf cohomology on the big étale site.

For your second question, I think this example shows that one approach is to look for a comparison like in the second theorem above betweeen $\mathrm{BG}$ and $\mathrm{BG}^\mathrm{top}$, and then to get information about $\mathrm{BG}^\mathrm{top}$ somehow. Even in the case above, though, Weibel's calculation looks difficult.


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